Line segment TS is tangent to circle O at point N. Circle O is shown. Line segment Q N goes from one side of the circle to the other side. Tangent T S intersects the circle at point N. Point P is on the circle between points Q and N. Point R is on the circle between points Q and N. Angle Q N T is 74 degrees.

If the measure of Angle Q N T is 74°, what is the measure of Arc Q P N?

37°
74°
148°
212°\

  148°

Step-by-step explanation:

The measure of the intercepted arc QN is twice the measure of inscribed angle QNT.

  arc QN = 2(74°) = 148°

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Comment on the question and answer

Your description "on the circle between points Q and N" is ambiguous. You used the same description for both points P and R. The interpretation we used is shown in the attachment. If point P is on the long arc NQ, then the measure of arc QPN will be the difference between 148° and 360°, hence 212°. You need to choose the answer that matches the diagram you have.

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We call angle QNT an "inscribed angle" because it is a degenerate case of an inscribed angle. The usual case has the vertex of the angle separate from the ends of the arc it intercepts. In the case of a tangent meeting a chord, the vertex is coincident with one of the ends of the intercepted arc. The relation between angle measure and arc measure remains the same: 1 : 2.